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In 1968, François Clemmons was a struggling opera student singing in a church choir when an unassuming middle-aged man approached him about appearing on a new television program for kids. Clemmons was skeptical. He had never heard of “Mister Rogers’ Neighborhood,” then in its first year, and the man, Fred Rogers, was almost suspiciously nice. Plus, the role he offered Clemmons was a cop.

“I come from the ghetto!” the now-73-year-old Clemmons tells The Post. As a black man, he adds, “I always thought of the police as being the most dangerous person in the neighborhood.”

Good thing he changed his mind.

A new documentary, “Won’t You Be My Neighbor?” (out Friday), focuses on the radical, revolutionary spirit of Mr. Rogers, who died in 2003. But Clemmons — who ended up playing the beloved Officer Clemmons on the show for 25 years — is the film’s unexpected star. A gregarious, now openly gay man, he was integral to making Mr. Rogers and his neighborhood so progressive for its time.

In an iconic 1969 episode, for example, Mr. Rogers and Officer Clemmons wade together in a kiddie pool, which aired in the midst of protests against segregated pools.

“François opened Fred’s eyes to a lot,” says Morgan Neville, who directed the doc. “He grew up in a broken home, he grew up poor, he grew up black and he grew up gay. And he talked to Fred about all those things.”

Clemmons was born in 1945 and raised in Youngstown, Ohio, by a family of Baptist choir singers. His childhood wasn’t a happy one.

“My parents fought,” he says. Both his father and his stepfather were abusive. “There was a lot of violence in my youth, and I was taken out of my home several times.”

Clemmons didn’t tell Rogers about his sexuality — until he was spotted at a gay club a few months into his tenure and word got out to the “Mister Rogers” brass.

“We talked about what it would mean to my role on the show, because this country would not tolerate an openly gay person on a children’s television program,” he says.

Rogers urged Clemmons to stay in the closet, and, in 1968, he even married a woman as cover. The union lasted only till 1974.

Hurtful as it was, Clemmons, who now lives Middlebury, Vt., says, “The experience was a blessing in disguise, because it allowed me to [eventually] say, ‘I am gay.’” He publicly came out immediately after his divorce. Though Officer Clemmons’ sexuality was never addressed on the show, Rogers aired episodes about the AIDS crisis and made a point of inviting openly gay guests on the program.

“[Rogers] changed,” says Nicholas Ma, a producer of the film who had appeared on “Mister Rogers’ Neighborhood” as a child with his father, cellist Yo-Yo Ma. And, he adds, Rogers’ and Clemmons’ relationship illustrates the two main tenants of the show: love and forgiveness.

“It’s so easy to believe that people are born accepting, but we have to work at it.”